Less fighting than expected experiments with wars of attrition and allpay auctions


 Matilda Neal
 1 years ago
 Views:
Transcription
1 Less fighting than expected experiments with wars of attrition and allpay auctions Hannah Hörisch Oliver Kirchkamp 27th June 2009 Abstract While allpay auctions are experimentally well researched, we do not have much laboratory evidence on wars of attrition. This paper tries to fill this gap. Technically, there are only few differences between wars of attrition and allpay auctions. Behaviourally, however, we find striking differences: As many studies, our experiment finds overbidding in allpay auctions. In contrast, in wars of attrition we observe systematic underbidding. We study bids and expenditure in different experimental frames and matching procedures and tie in with the literature on stepwise linear bidding functions. Keywords: War of attrition, dynamic bidding, allpay auction, stabilisation, volunteer s dilemma, experiment JEL classification C72, C92, D44, E62, H30 Universität Bonn University of Jena; School of Economics; CarlZeißStr. 3; Jena; Germany. Fax: , Phone: , An earlier version of this paper circulated under the title Why are stabilisations delayed an experiment with an application to all pay auctions. We thank Sarah Volk for valuable support during the preparation of this paper and the experiment, Radosveta IvanovaStenzel, Hartmut Kliemt, and two anonymous referee for helpful comments on earlier versions, and Urs Fischbacher (2007) for ztree. Financial support from the Deutsche Forschungsgemeinschaft through SFB 504 is gratefully acknowledged.
2 1 Introduction Numerous experiments explore models of a static conflict or competition such as firstprice allpay auctions and rentseeking contests. The laboratory evidence documents that human decision makers tend to overbid and make investments that are even higher than the already high and socially wasteful equilibrium values. This paper investigates which of the properties of the static firstprice allpay auction carry over to the war of attrition, i.e. a secondprice allpay auction. As a benchmark we replicate the standard firstprice allpay auction experiment in a situation with a single prize and two bidders. As expected, we observe some overbidding. Similar to Müller and Schotter (2007) we also find bifurcated bidding functions, i.e. stepwise linear bidding functions. We extend the experimental literature by investigating a war of attrition, which we implement both as a dynamic and as a static bidding process. In contrast to the literature on allpay auction we find underbidding, in particular in the dynamic implementation of the war of attrition. Furthermore, we do not find bifurcated bidding functions in the war of attrition. A war of attrition can be interpreted as a fight between two players about a prize. For each second during which players are fighting both players incur a cost. As soon as one player gives up, the conflict ends and the other player earns the prize. Both players have to make their investments. Due to its dynamic character the war of attrition has been used as a model of a wide range of problems: Arms races (Zimin and Ivanilov, 1968), fights between animals (Bishop, Canning, and Smith, 1978; Riley, 1980), the voluntary provision of public goods (Bliss and Nalebuff, 1984; Bilodeau and Slivinski, 1996), competition between firms (Ghemawat and Nalebuff, 1985; Fudenberg and Tirole, 1986; Roth, 1996), the settlement of strikes (Kennan and Wilson, 1989; Card and Olson, 1995), fiscal and political stabilisations (Alesina and Drazen, 1991; White, 1995), the timing of exploratory oil drilling (Hendricks and Porter, 1996), and many others are all wars of attrition. Wars of attrition with two players are isomorphic to (static) secondprice allpay auctions. In a secondprice allpay auction with two bidders both bidders submit a bid (this bid is equivalent to the amount of time bidders are willing to 1
3 hold out during the war of attrition), the bidder with the highest bid wins the prize, and both bidders pay the amount of the second highest bid. Also in the war of attrition the winner does not have to pay more once the loser gave up, i.e. pays effectively the second highest bid. ½ Wars of attrition are also similar to firstprice allpay auctions. A firstprice allpay auction (which we will simply call allpay auction in the following) is a static procedure where bidders commit to bids or investments previous to the game. The highest bid wins and all bidders pay their own bid. While there are many experiments on allpay auctions there are surprisingly few on wars of attrition. Our paper tries to fill this gap. We will show that behaviour is quite different under the two institutions. Table 1 on page 3 provides an overview of the experimental literature. Experiments with allpay auctions have been done by Davis and Reilly (1998), Potters, de Vries, and van Winden (1998), Barut, Kovenock, and Noussair (2002), Gneezy and Smorodinsky (2006), and Müller and Schotter (2007). Four of these studies find a significant amount of overbidding while one (Potters, de Vries, and van Winden, 1998) finds no significant deviation from equilibrium. A famous variant of an allpay auction is the dollar auction game (Shubik, 1971). Again, overbidding is a standard finding in laboratory experiments. Müller and Schotter (2007) observe what they call bifurcation of bidding functions in experiments with allpay auctions. Compared with equilibrium bids bids are too small if bidding is expensive, and too large if bidding is cheap. Bidding functions can be approximated with the help of a stepwise linear function similar to the one in figure 1. In section 4.4 we will try to replicate Müller and Schotter s results and check whether these bifurcated bidding functions can also be found in a war of attrition. Another related auction format is a rentseeking contest. As in allpay auctions also in rentseeking contests all players pay their own bid. In contrast to the allpay auction the highest bidder in the rentseeking contest does not win the auction with certainty. Instead, the probability of winning is only an increasing function of the bid. ¾ Rentseeking contests have been introduced by Bishop, Canning, and ½ With more than two players the two formats are no longer isomorphic, though, weakly equivalent. In the experiment we present below we will concentrate on the situation with two players. ¾ In Krishna and Morgan (1997) only allpay auctions in which the highest bidder obtains the prize with certainty are called allpay auctions. 2
4 Table 1 Comparison with other experiments 3 institution bidding procedure asymmetric information about Millner and Pratt (1989) rentseeking static overbidding Shogren and Baik (1991) rentseeking, framed in expected payoffs number of contestants number of prizes repetitions in a group periods static result close to equilibrium Davis and Reilly (1998) rentseeking firstprice static overbidding Potters, de Vries, and van Winden (1998) rentseeking overbidding in the static firstprice rentseeking case Vogt, Weimann, and Yang (2002) rentseeking sequential efficient equilibrium selected Barut, Kovenock, and Noussair (2002) firstprice static valuation 6 2 or or 50 overbidding Anderson and Stafford (2003) rentseeking cost, no. of static with entryfee contestants overbidding Schmidt, Shupp, Swope, and Cardigan (2004) rentseeking static overbidding Schmidt, Shupp, and Walker (2004) rentseeking static 4 1,3, 1 1 underbidding Bilodeau, Childs, and Mestelman (2004) volunteer s dilemma dynamic overbidding Öncüler and Croson (2005) risky rentseeking static 2, overbidding Gneezy and Smorodinsky (2006) firstprice static 4,8, overbidding Abbink, Brandts, Herrmann, and Orzen (2007) rentseeking static overbidding Müller and Schotter (2007) firstprice static cost 4 1,2 1,50 50 overbidding, bifurcation underbidding our experiment secondprice dynamic in the sequential format, no cost 2 1 1,6 24 firstprice static bifurcation
5 Figure 1 A bifurcated bidding function (solid) and an equilibrium bidding function (dashed) β(c) Firstprice allpay β A (c i ) c c c i Smith (1978) and Tullock (1980). Possibly the first test of Tullock s model in the laboratory is an experiment by Millner and Pratt (1989). Their, perhaps disappointing, result is that participants make substantially larger bids, i.e. socially wasteful investments, in the rentseeking contest than they should in equilibrium. Several other experiments with rentseeking contests followed. Seven of the ten studies of rentseeking contests find a significant amount of overbidding, two studies (Shogren and Baik, 1991; Vogt, Weimann, and Yang, 2002) find no significant deviation from equilibrium, and only one (Schmidt, Shupp, and Walker, 2004) finds significant underbidding in rentseeking contests. A generalised war of attrition, or a volunteer s dilemma, is an nth price allpay auction with n 1 > 1 prizes. A simple version of such a game has been studied experimentally by Diekmann (1993). Participants decide whether to provide a public good. Each player can only choose between two different bids, a low bid (waiting for somebody else to provide the public good) or a high bid (provision of the public good). Bilodeau, Childs, and Mestelman (2004) study experiments of a sequential version of this game where players are fully informed about each other s bidding costs and find overbidding. Following Tullock allpay auctions can be seen as a special case of rentseeking contests. Shogren and Baik (1991), Davis and Reilly (1998), Potters, de Vries, and van Winden (1998), Vogt, Weimann, and Yang (2002), Anderson and Stafford (2003), Schmidt, Shupp, Swope, and Cardigan (2004), Schmidt, Shupp, and Walker (2004), Öncüler and Croson (2005), Abbink, Brandts, Herrmann, and Orzen (2007). See Bulow and Klemperer (1999) for a derivation of the equilibrium of such a game. 4
6 In sum, most of the experiments with variants of allpay auctions find systematic overbidding, i.e. bids that are higher than the risk neutral equilibrium. Actually, even in regular (winneronlypays) secondprice and firstprice auctions overbidding is a common phenomenon, although to a lesser degree. If we extrapolate from allpay auctions and rentseeking contests, we should expect some overbidding in wars of attrition, too. We will proceed as follows: In section 2 we derive the equilibrium bidding functions. In section 3 we discuss the experimental procedure and present our setup. Section 4 presents the results and section 5 concludes. 2 Hypotheses and Equilibrium predictions 2.1 Equilibrium in the war of attrition In this section we derive the risk neutral Bayesian Nash equilibrium for the war of attrition with two bidders, i and j. As noted above, wars of attrition with two players are isomorphic to secondprice allpay auctions. We will call P the prize from winning the war of attrition. In the auction bidders make bids b by staying for up to b seconds in the auction unless the opponent leaves the auction earlier. If bidder i is staying for b seconds in the auction this bidder incurs a cost b c i, where c i is private information of bidder i and not known to bidder j. The other bidder only knows that c i is drawn from a uniform distribution over [c c] where c is positive and should not be too small compared to c to ensure that the participation constraint is fulfilled. We follow a standard approach and assume that the opponent of bidder i, bidder j with cost per second c j, uses a decreasing bidding function β W e (c) with inverse β W( 1) e ( ). Bidder i does not know c j, but can calculate the expected utility Í i, given that i s bidding cost per second are c i and bidder i bids up to b seconds. If bidders maximise a utility function u(x) See Cox, Smith, and Walker (1988); Kagel, Harstad, and Levin (1987); Kagel and Levin (1993). 5
7 the expected utility is β W e ( 1) (b) c Í i = u( b c i ) f(c j ) dc j + u ( P β W e (c ) j) c i f(cj ) dc j c β }{{} W ( 1) e (b) }{{} bidder i does not win and pays the own bid b bidder i wins P and pays the second highest bid β W e º (1) We will first analyse the risk neutral case. Thus, we can replace u(x) by x, take the first derivative Í i / b and substitute b = β W e (c i) to obtain the first order condition c i ci c f(c j )dc j P f(c i) β W e (c i ) = 0 º (2) If c i is uniformly distributed over [c c] then (2) can be simplified: β W e P (ci ) = c i (c i c) (3) which, with the restriction β W e ( c) = 0, yields the candidate equilibrium bidding function: β W e (c i) = P c ÐÒ ( c c)c i (c i c) c To check whether the individual participation constraint is fulfilled we substitute (4) into (1) and obtain Í i = P c c i + ÐÒ c i c c c (4) º (5) Hence, as long as c is not too small the expected utility from participating is positive for all c i [c c]. This is the case for all combinations of parameters we use in our experiment (see table 2 below). Hence, the expression given by equation (4) describes an equilibrium bidding function. From equation (4) we see that a bidder with a very small c i is ready to make a potentially very high bid. It might even be that β W e (c i) c i > P. Since we are in a war of attrition, a bidder with a very small c i will pay the bid of the opponent with the larger c i which is much smaller than β W e (c i ) most of the time. Therefore, as we see in equation (5), the expected utility, ex ante, is still positive. We should also note that, during the bidding process, our bidder does not necessarily want to stop when b c i = P. During the war of attrition the current bid b c i is already 6
8 Figure 2 Equilibrium bidding functions risk neutral bids constant absolute risk aversion war of attrition allpay auction β e β e β e war r = of attrition firstprice allpay c i r r = = c c c c c The left diagram shows risk neutral equilibrium bidding functions for the war of attrition (solid line) and allpay auction (dashed lines). The diagram in the middle shows equilibrium bidding functions for the war of attrition for bidders with constant absolute risk aversion and risk aversion parameters r = 2, r = 6, and r = 20. The diagram on the right shows similar equilibrium bidding functions for the allpay auction c i r r = r = = c c i a sunk cost at any point in time. Only the marginal increment of the bid and the thereby achieved increase in probability to obtain P are relevant. Hence, while it might seem strange that bidders make bids potentially larger than P, (with our parameters) they will make a positive profit ex ante, and they should not care about their sunk investment ex interim. An example of the equilibrium bidding function is shown as the solid line in the left diagram in figure 2. The expected expenditure is R W e = c c 2.2 Equilibrium in the allpay auction ( ) β W e (c i) 2 c i c c c 1 + ÐÒ c ( c c) dc c 2 i = P (6) ( c c) 2 Similar to the derivation for the war of attrition in equations (1) to (5) we can find the equilibrium bidding function for the allpay auction. We assume that bidder j with cost c j uses a decreasing bidding function β A e (c) with inverse β A( 1) e ( ). The Many auction theorists call the sum of all bids revenue and use the letter R to denote revenue. Since in many applications of wars of attrition there is no seller and no revenue we use the term expenditure in this paper. 7
9 expected utility of bidder i with per period cost c i who bids up to b periods is β A e ( 1) (b) c Í i = u( b c i ) f(c j ) dc j + u (P b c i ) f(c j ) dc j c β }{{} A ( 1) e (b) }{{} bidder i does not win and pays the own bid b bidder i wins P and pays the own bid b º (7) Again, we will first consider the risk neutral case u(x) = x. We take the first derivative Í i / b and substitute b = β A e (c i) to obtain the first order condition c i ci c c f(c j )dc j c i f(c j )dc j f(c i) c i β A e (c i ) P = 0 º (8) With c i distributed uniformly over [c c] we can simplify (8) to β A e P (ci ) = c i ( c c) (9) which, with the restriction β A e ( c) = 0, yields the candidate equilibrium bidding function: β A e (c i) = P c c ÐÒ c c i º (10) To check whether the individual participation constraint is fulfilled we substitute (10) into (7) and obtain Í i = P ( c c i)( c c) ÐÒ c c i c c º (11) Again, this expression is positive for all c i [c c] as long as c is not too small. For our parameters this is always the case, hence, the expression given by equation (10) describes an equilibrium bidding function. An example of the equilibrium bidding function is shown as a dashed line in the left part of figure 2. As long as c i (c c) we always have β A e (c i) < β W e (c i). The expected expenditure of the allpay auction is R A e = c c ( ) β A e (c c c 1 + ÐÒ i) c c c dc c i = P ( c c) 2 º (12) 8
10 Comparing with equation (6) we see that the expected expenditure in the two auction formats is the same, R W e = RA e. Krishna and Morgan (1997) point out that this need not be the case if bidding costs are not distributed independently. They show that generally the expenditure in the war of attrition is larger than or equal to expenditure in the allpay auction. 2.3 Risk aversion In the previous two sections we have derived equilibrium bidding functions under the assumption of risk neutrality. We know from other auction experiments that bidders behaviour can often be well explained by risk aversion. In this section we derive equilibrium bidding functions for risk averse bidders. While we can not find a closed form solution for general utility functions, it is possible to follow the above steps for specific utility functions. In our context, constant relative risk aversion u(x) = x 1 r is not very meaningful since equilibrium payoffs can be positive and negative. Thus, utility is sometimes real and sometimes complex, which is difficult to interpret. Here we consider only the case of constant absolute risk aversion u(x) = e r x. War of attrition function is With constant absolute risk aversion r the equilibrium bidding β WR e (c i ) = c c ) c c (1 e rp ÐÒ ( c c)c i rc (c i c) c (13) and the expected expenditure is R WR e = 1 e rp ( ( c c r ( c c) c c 1 + ÐÒ c )) º (14) c In the risk neutral limit r = 0 these expressions coincide with their risk neutral counterparts given by (4) and (6). For all positive values of r bids and expenditures are decreasing in r and smaller than the risk neutral values. 9
11 Firstprice allpay auction β AR e = The equilibrium bidding function is ) c c ( c c) (1 e rp ) ÐÒ c c r ( c e rp c ) c c (( c c i ) e rp + c i c c (15) ( c c) c i and the expected expenditure is R AR e = (c ÐÒ cc + rp )e rp c c c ÐÒ c c r (e rp c c c c) º (16) In the risk neutral limit r = 0 these expressions coincide with their risk neutral counterparts given by (10) and (12). For positive values of r bids are smaller if c i is sufficiently large. This is consistent with the observation of Fibich, Gavious, and Sela (2006). In their paper risk is modelled in a different way, as weak risk aversion, which is a small perturbation of the risk neutral utility function. They also find that with increasing risk their low type, in our model the bidder with a high c i, bids less the more risk averse bidders become. However, the high type, in our model the bidder with a low c i, bids more the larger the amount of risk aversion. The intuition given by Fibich, Gavious, and Sela also applies in our case: Bidders with a high c i will, most likely, not win the auction and just lose their bid. Thus, making a high bid is risky for them. The more risk averse bidders are, the lower is the equilibrium bid. Bidders with a small c i have good chances to win. Making a slightly too small bid can be very risky for them. Thus, the more risk averse bidders become, the higher are the equilibrium bids for bidders with a low c i. The right diagram in figure 2 shows bidding functions for different values of risk aversion r. We see that for most values of c i risk averse bids are below the risk neutral bid. However, if c i is very small, risk averse bids are higher than the risk neutral bid. The expenditure is decreasing in r and smaller than the risk neutral expenditure. For the parameters we are using R AR e decreases more slowly than R WR e, i.e. with risk averse bidders the expected expenditure is larger with allpay auctions. 10
12 3 Experimental setups We will use table 1 to discuss our setup. Institution: Most experiments summarised in table 1 have been done with allpay auctions and rentseeking contests. There is also one experiment with a volunteer s dilemma. In this paper we concentrate on wars of attrition. We use allpay auctions as a reference point. With our experiment we want to find out whether the seemingly small differences between the two auction formats affect bidding behaviour and bidding anomalies. Another game that is similar to a war of attrition is the sequential volunteer s dilemma study by Bilodeau, Childs, and Mestelman (2004). In this game players have symmetric information about each other s bidding cost. In the only subgame perfect equilibrium of their game the bidding process ends in the first round with a bid of zero, i.e. in equilibrium one does not observe dynamically increasing bids. Bilodeau, Childs, and Mestelman observe substantially higher bids, i.e. again overbidding. We depart from Bilodeau, Childs, and Mestelman in analysing a situation with asymmetric information. The mutual bidding cost is only revealed at the end of the game, i.e. players do not know ex ante who is the weakest bidder. Bidding procedure: Many other auction experiments use a static bidding procedure. In this paper we will compare two procedures: A static and a dynamic procedure. With the static procedure players simply fill in a number on the screen and the computer immediately determines the winner and gains and losses from the game. With the dynamic procedure participants see their bid and the time like an ascending clock on the screen and can, similar to the procedure in the Dutch auction, decide to stop the clock. Unlike in the Dutch auction, the person who stops the clock is the loser of the auction. With two bidders the two processes are isomorphic, but behaviourally there could be a difference. The dynamic procedure applies naturally in the context of the war of attrition. However, if we find Note that, as in the Dutch auction, bidders in a twoplayer war of attrition do not learn anything unexpected until the auction ends. If a bidder finds it optimal to hold out until a bid b ex ante, no relevant new information is gained ex interim. 11
13 behavioural differences between allpay auctions and wars of attrition it will be interesting to know more about the reasons for these differences: Does behaviour differ as the result of a change from a static to a dynamic procedure, or as the result of a change in the strategic situation? Hence, we will study the war of attrition both with a dynamic and a static procedure. Asymmetric information: Most of the early experiments with allpay auctions and rentseeking contests use symmetric information of players about all aspects of the game. This symmetry simplifies the experiment considerably. Only some of the more recent experiments (see table 1) introduce asymmetric information. In our introduction we mentioned a couple of examples for war of attritions. In these examples the bidding cost, i.e. the strength of the army or the animal or the stength of the R&D department of a firm, the waiting cost during the settlement of a strike, etc. is neither perfectly known to everybody nor is it the same for all contestants. More technically, the rather abstract case of perfect symmetry rules out equilibria in pure strategies. To avoid mixed equilibria as a benchmark solution and to be closer to the applications we have in mind we will assume that bidding costs are private information and are uniformly distributed over some interval [c c]. Most of the treatments we did were based on intervals with c/ c = 1/2 with c between 3º6 and 5º2 (see appendix A). After normalising the bids we did not find any significant differences between these intervals. Hence, we decided to pool treatments with c/ c = 1/2 but different c. Number of contestants and prizes: Most of the experiments that we summarise in table 1 have two bidders. A setup with two bidders is particularly attractive for a comparison of a dynamic with a static game, since with two bidders the static and the dynamic game are isomorphic. We will, hence, have two bidders and one prize in our experiment. In this context we should mention Anderson, Goeree, and Holt (1998) who develop a model of boundedly rational bidders which relates overbidding to the number of bidders. The more bidders there are the more overbidding we should observe in this model. With two bidders we have the smallest possible number 12
14 of bidders, thus we should not expect too much overbidding. Since we are not interested in the absolute level of overbidding but in a comparison of bidding behaviour in different auction formats, the number of bidders is not essential for us. Repetitions: The Bayesian Nash equilibria, which we present in sections 2.1, 2.2, and 2.3, are the equilibria of a game that is played once or, by backward induction, equilibria of a game that is played finitely many times. We will study both situations: a sequence of one shot games with random matching of players after each round, but also games with random matching of players only after a fixed number of repetitions. While the one shot game constitutes a simple benchmark, we find the treatment with repetitions more interesting. Players who have fought over the allocation of a resource in the past will do this again. Regardless of whether we interpret the war of attrition as an arms race, as the provision of public goods, the competition between firms, the settlement of strikes, or as political stabilisation wars of attrition tend to repeat. Hence, most of our experiments are based on the finitely repeated war of attrition. The repeated situation is not only closer to the conflict we actually want to model, but studying repeated wars of attrition also has a technical advantage in the laboratory since the waiting time for other participants is considerably reduced. Nevertheless, we also have one treatment with random rematching of players after each round in sections 4.2 and 4.3. Implementation in the lab: All treatments of the experiment were implemented with the software ztree (Fischbacher, 2007) and carried out at the experimental laboratory of the SFB 504 at the University of Mannheim. In our experiment we want to study the impact of the number of repetitions, the role of static versus dynamic bidding, and the difference between a war of attrition and an allpay auction. Our baseline treatment is a (dynamic) war of attrition with 6 repetitions and cost intervals with 2c = c. The other four treatments are variants of the baseline treatment in which only one factor is changed. Table 2 lists the different treatments. Parameters for each session are given in section A of the appendix. In Since bids can have a large variance some wars of attrition will end quickly while others last for a long time. If players are rematched in each period, most players will have to wait most of the time. Otherwise these different bids average out, which speeds up the experiment considerably. 13
15 Table 2 Treatments c/ c repetitions static firstprice participants sessions β e < b The last column shows the relative fraction of bids b that are larger than the equilibrium bid β e. Figure 3 The bidding interface in dynamic treatment round: 2 of 24 remaining time [sec]: 3596 The value of the prize is 100 The cost of the other bidder is between 2.2 and 4.4 per second Your cost is 3.59 per second You are now bidding the following number of seconds for the prize: 4.00 You have, hence, bid the following amount in this auction: To leave the auction, press the bottom right button I stop bidding the remainder of this section we describe how the treatments were implemented. In the treatments with dynamic bidding groups of typically 10 to 14 participants read instructions (see section B in the appendix), answer computerised control questions to check whether they understand the experiment, and are matched randomly in pairs to bid for a prize. During the bidding process participants see information similar to the one shown in figure 3. ½¼ The number of seconds and the bid are updated every second. As soon as one bidder stops bidding the other is declared winner of the auction. In the static treatment, the screen look looks like the one in figure 4. In ½¼ For technical reasons the top right corner of the screen in the experimental interface also showed a remaining time slowly counting down. We set this time to an exorbitantly high value, so it never became binding in any of our experiments. In the unlikely event of remaining time actually counting down to zero the text would change to please make your decision now but bidders would still be able to increase their bids. This, however, never happened in any of our 14
16 Figure 4 The bidding interface in the static treatment round: 2 of 24 remaining time [sec]: 45 The value of the winner s prize is 100 The cost of the other bidder is between 2.2 and 4.4 per second Your cost in this round is 3.59 per second Please enter the amount of seconds or total cost which you are ready to bid Maximal cost to bid to seconds to cost maximal number of seconds 4.00 Continue Figure 5 Feedback given at the end of a round round: 2 of 24 remaining time [sec]: 17 The other bidder has won the auction auction your other length of your other your new balance cost per bidder s the auction cost bidder s profit of second cost per in (total) cost in this your ac second seconds (total) auction count contrast to the dynamic procedure (figure 3) players make decisions before the actual bidding begins. They either fix a highest cost or a highest number of seconds up to which they are willing to bid. The interface in the experiment automatically converts cost into seconds and vice versa. Once players have made their choice the actual bidding process completes instantaneously. ½½ In both the dynamic and the static treatment participants get feedback similar to the one shown in figure 5. They are asked to copy this information manually to a table on their desk. Some of the feedback information, such as the other bidder s cost, might not always be available to bidders in a natural context. However, during our pilot experiments we saw that presenting the information in a symmetric experiments. ½½ As long as we look at bidders who make a positive bid participants spend on average about the same amount of time with the dynamic and the static decision screen (33.41 seconds in the dynamic treatment and seconds in the static treatment). 15
17 Figure 6 Cumulative distribution of participants total payoffs in the experiment Fn(x) payoff / [Euro] way helps participants to understand the nature of the game. Depending on the treatment participants are either rematched every round or every six rounds to a new random partner. This procedure is repeated until the 24th round. At the end of the experiment participants complete a questionnaire, and receive their earnings from the experiment in sealed envelopes. Sessions last for about 75 minutes. The cumulative distribution of payoffs at the end of the experiment is shown in figure 6. 4 Results 4.1 Bidding Equations (4) and (10) describe equilibrium bids β W e and βa e in the war of attrition and allpay auction, respectively. The higher the individual cost c i the earlier a participant will give up. The last column of table 2 shows the relative frequency of bids b that are larger than equilibrium bids β e for the different treatments. We can already see that overbidding is more frequent in the allpay auction and in the static version of the war of attrition while in all dynamic versions of the war of attrition the majority of bids are smaller than equilibrium bids. Figure 7 shows scatterplots of bids in the war of attrition and in the allpay auction. Let us first have a look at the left graph in figure 7. Each dot corresponds to one bid in the dynamic war of attrition. We also show contour lines of a kernel density estimate. The solid line indicates the equilibrium bid (equation 16
18 Figure 7 Scatterplots of bids war of attrition (dynamic) war of attrition (static) all pay auction b c equilibrium bid P/c 20 b c equilibrium bid P/c b c equilibrium bid P/c c c c c c c The figure shows bids and contour lines of a kernel density estimate from all treatments where c = 2c. Out of the 1788 individual bids in the left graph 64 are out of the range of the graph. Out of the 600 individual bids in the center graph 9 are out of the range of the graph. The graph for the allpay auction includes also the bids of the winners. We use R s kde procedure with default parameters for the kernel density estimate. (4)). Generally, bidders with a large cost make smaller bids which is in line with the equilibrium prediction. We also see that there is a substantial variance. Furthermore, we find that many bids for the smaller values of c in the war of attrition concentrate around P/c i (dashed line). These bidders leave the auction when they have exactly spent the value of the prize. It also seems that bids follow a different pattern for small and for large values of c. We will come back to this point in section 4.4 when we talk about bifurcated bidding functions. In section 4.4 we will also discuss why the graphs in figure 7 look so differently. Meanwhile, we will provide a more formal comparison of the treatments in the next sections. 4.2 Comparison with equilibrium bids We use an interval regression to estimate the following equation: b i t = β e i (1 + β Û Ö d Û Ö + β Ø Ø d Ø Ø + β ÖÒ d ÖÒ + β ÐÐÔ d ÐÐÔ ) + ν i + u i t (17) where β e i is the equilibrium bid from equation (4) or (10) and b i t is the actual bid. Since we do not observe the winner s bid in the dynamic war of attrition we use an interval regression (see Tobin, 1958; Amemiya, 1973, 1984; Quing and Pierce, 1994). d Û Ö is a dummy that is one in the dynamic war of attrition with 17
19 Table 3 Interval regression random effects estimation of equation (17) β σ z P >t ±ÓÒ º ÒØ ÖÚ Ð β Û Ö º4245 º º415 0º000 º46526 º38375 β Ø Ø º25721 º º609 0º000 º33349 º18093 β ÖÒ º19658 º º209 0º000 º28813 º10504 β ÐÐÔ º68562 º º508 0º000 º47913 º89211 The estimation includes only experiments with c/c = 2. repeated interaction, d Ø Ø is one in the static war of attrition, d ÖÒ is one in the war of attrition in which players are randomly rematched after each period, and d ÐÐÔ is one in the allpay auction. ν i is a random effect for each subject, u i t is the noise term for each individual choice. If bidders follow the equilibrium bidding function then we should estimate β Û Ö = β Ø Ø = β ÖÒ = β ÐÐÔ = 0. ½¾ Table 3 presents an interval regression random effects model with a random effect for each participant. ½ In the estimation of equation (17) we observe the following: In the allpay auction there is a significant amount of overbidding (about 69%). This is what we should expect from previous experiments. In all other situations, the dynamic war of attrition, the static war of attrition, and the war of attrition with random rematching we find underbidding (between 20% and 42% depending on the treatment). In particular, underbidding in the war of attrition is always significant. We also ran a regression with an additional dummy which was one if the participant won the auction in the previous period and zero otherwise. The coefficient of such a dummy is small (0.045) and not significant (pvalue of 0.113). Estimates of the ½¾ The ratio between risk averse equilibrium bids and risk neutral bids is (1 e 200 r )/200r for the war of attrition and (200r + ÐÒ2 e 200 r ÐÒ2)/(200 (2 e 200 r ) r (1 ÐÒ2)) for the allpay auction. Hence, when we compare the allpay auction with the war of attrition then we can t simply compare levels of these ratios (or levels of estimated coefficients). We can still compare bids in the experiment with risk neutral bids. If we observe a significant amount of underbidding (compared with risk neutral bids) in the war of attrition and a significant amount of overbidding in the allpay auction then no degree of risk aversion can be consistent with both observations. ½ A model with random effects for each session of the experiment yields very similar coefficients and pvalues. A model with fixed effects for participants can not be used here since participants are fully nested in treatments. 18
20 Figure 8 Estimation of equation (17) for different quarters of the experiment β war static rand. allp first second third last Periods [ Quarters of the Experiment ] other coefficients (and their pvalues) do not change substantially, we therefore do not provide an extra table for this estimation. To see whether players systematically change their behaviour during the experiment and perhaps converge to the Bayesian Nash equilibrium we divide the periods in the experiment into four parts of equal size, the first, the second, the third, and the last quarter of the experiment. We estimate equation (17) separately for each quarter. Results are shown in figure 8. We see that for all auction formats bids decrease over time in a similar way. The distance between the war of attrition and the allpay auction remains similar during the course of the experiment. To summarise this subsection: While the experimental literature on allpay auctions and rentseeking contests typically finds overbidding our results document underbidding in the warofattrition. ½ 4.3 Comparison of expenditures Based on the equilibrium expenditures given by equations (6) and (12) and for our distribution of bidding cost we should expect that the level of expenditure is the ½ Among experimental papers on allpay auctions and rentseeking contests only Schmidt, Shupp, and Walker (2004) find significant underbidding in three different types of rentseeking contests (a random single prize, a random multiple prize and a deterministic proportionate prize contest). The results of Schmidt, Shupp, and Walker (2004) are hard to compare to ours since their setup differs from ours along many dimensions, e.g. the type of game, its static character, the number of contestants. Additionally, in Schmidt, Shupp, and Walker (2004) each participant played the three different contests but each contest was played oneshot, each participant has the same bidding costs and there is no asymmetric information concerning bidding costs. 19
21 Table 4 Expenditure: Results of estimating equation (18) β σ t p value 95% conf interval β Û Ö β Ø Ø β ÖÒ β ÐÐÔ Random effects regression with a random effect for subject and one for matching group. The estimation includes only experiments with c/c = 2. same under the war of attrition and the allpay auction. For other distributions of the bidding cost this need not be the case. One can show that in equilibrium expected expenditures in the war of attrition are never smaller than in the allpay auction, they can only be larger (see Krishna and Morgan, 1997). In the experiment we have seen that bids are smaller than equilibrium bids in the war of attrition and larger in the allpay auction. What can we now say about expenditure? Average overbidding in the allpay auction does not necessarily imply higher expenditures, at least not if the amount of overbidding depends on the bidding cost. Similarly, average underbidding in the war of attrition does not need to imply lower than equilibrium expenditures. To compare expenditures in the experiment we estimate the following random effects model R i t = R e i t (1 + β Û Öd Û Ö + β Ø Ø d Ø Ø + β ÖÒ d ÖÒ + β ÐÐÔ d ÐÐÔ ) + ν i + ν s + u i t (18) where R i t is the actual expenditure obtained in the auction and R e i t the expost expected expenditure in equilibrium. d Û Ö, d Ø Ø, d ÖÒ, d ÐÐÔ, ν i, and u i t have the same interpretation as in equation (17). ν s is a random effect for each session. If all bidders follow the Bayesian Nash equilibrium bidding function we should have β Û Ö = β ÐÐÔ = β ÖÒ = β Ø Ø = 0. If the coefficients are negative the expenditure is smaller than in equilibrium. If the coefficients are positive then expenditure is larger than in equilibrium. Results of the estimation are shown in table 4. ½ We observe the following: For all variants of the war of attrition that we studied (dynamic or static ½ Outliers have been eliminated using Hadi s method (Hadi, 1992, 1994). 20
22 bidding, random or repeated matching) the estimated coefficients are significantly negative, i.e. the expenditure in the war of attrition is smaller than the expenditure in equilibrium. Expenditures in the allpay auction are larger than in the war of attrition ½ but significantly smaller than equilibrium expenditures. Thus, while expenditures in the war of attrition and the allpay auction are still socially wasteful in our experiment their level is lower than predicted by equilibrium. This might look surprising since we found bids in the allpay auction to be larger than equilibrium. The next section might offer an answer to this puzzle. 4.4 Bifurcations in allpay auctions and in wars of attrition Let us have another look at figure 7. The three scatterplots show bids for the dynamic war of attrition, the static war of attrition, and the allpay auction. The figure suggests that, in particular for the allpay auction, bids can be divided into two distinct groups: bids for large values of c and bids for small values of c. The latter are high (and do not depend much on the specific value of c), the former are low (and, again, do not depend much on c). In an experiment with allpay auctions Müller and Schotter (2007) observe a similar phenomenon and call it bifurcation of bidding functions. If bidding is expensive, bidders underbid. If bidding is cheap, bidders overbid (see figure 1 on page 4). We want to find out whether this is a stable pattern that repeats in our experiments with wars of attrition, i.e. whether individuals with low costs fight longer than predicted by equilibrium and those with high cost stop fighting earlier. The left diagram in figure 7 already suggests that this phenomenon is less pronounced in the war of attrition. However, figure 7 only shows the aggregate distribution. In this section we will look at individual bidding functions. We follow the procedure suggested by Müller and Schotter. If bids can also be divided in the war of attrition, this procedure might be a sensible way to structure ½ A twosided (parametric bootstrap) test of β Û Ö = β ÐÐÔ yields a p = 0º
23 Figure 9 Stepwise approximation of bidding functions for 18 participants in the war of attrition estimated stepwise bidding function, loss, win The nine diagrams on the left are examples for bidders who can best be approximated with a descending step in the bidding function, the nine on the right are examples for bidders who can best be approximated with an ascending step in the bidding function. the data. Müller and Schotter estimate the following switching regression: b i (c) = { βe i c + β 0 i + u if c ^c i γ e i c + γ 0 i + u if c > ^c i º (19) We can use an OLS regression for the static war of attrition and the allpay auction. For the dynamic war of attrition we have to use an interval regression (Tobin, 1958; Amemiya, 1973, 1984). For each individual i we estimate coefficients ^β e i ^β 0 i ^γ e i ^γ 0 i. Also for each individual we choose the position of the step ^c i such that the likelihood of the interval regression is maximised. ½ Figure 9 shows several examples of individual bids and estimated individual bidding functions. The nine diagrams on the left show examples for bidders who can best be approximated with a descending step in the bidding function, the nine on the right are examples for bidders who can best be approximated with an ascending step in the bidding function. Müller and Schotter test their approach by comparing the residual sum of ½ For the interval regression we do not obtain convergence of the estimator for 2 of our 282 participants. To check overall convergence we did all our estimates twice, once with at most five iterations, and once with at most 25 iterations. The results are practically the same (we have looked at distributions of estimated coefficients like those presented in figure 10 and found that they are visually indistinguishable for 5 and for 25 iterations) so that we have no reason to believe that a larger number of iterations might change the results. 22
24 Figure 10 Bifurcation, the distribution of i Fn(x) dynamic war static war all pay i The graph shows the cumulative distribution of the step size i. For the static treatments i is given for an OLS approach, for the dynamic treatment i is shown for the interval regression. squares of the switching regression model (19) with the residual sum of squares of an alternative model (which contains the equilibrium bidding function as a special case). We did the same with our data and come to the same conclusion: the switching regression model has a better fit. We should keep in mind that the switching regression model permits all kinds of stepwise functions. Figure 9 suggests that in the war of attrition not all participants use bidding functions that are flat for small bidding costs, then decrease sharply, and then remain flat for high bidding costs. We have to ask ourselves which examples are more representative of bidding in the war of attrition: the ones in the left part of figure 9 or the ones in the right part? To answer this question we will look at the step size, i.e., the difference i = ^β e i^c i + ^β 0 i (^γ e i^c i + ^γ 0 i ) º (20) This difference is positive if participants have a decreasing step in their bidding function like in the left part of figure 9. Figure 10 shows the distribution of the step size. Let us first look at the dotted line which shows the distribution of the step size i for the allpay auction. As we should expect from Müller and Schotter, more than 50% (actually 59.1%) have an estimated positive stepsize. For the different treatments we test against i = 0 and report results in table 5. For the allpay auction and the static war of attrition we do not find i to 23
25 Table 5 Twosided tests against i = 0 i t P > t P Ò dynamic war of attrition (interval regression) static war of attrition (OLS) static allpay auction (OLS) tstatistics are derived using the HuberWhite method to correct for correlated responses from cluster samples. P Ò are p values of a binomial test. Figure 11 Approximating equilibrium bidding functions with stepwise linear functions b war of attrition allpay auction c c be significantly different from zero. i is actually significantly negative in the dynamic war of attrition where only 36.6% of participants seem to use a positive stepsize. How can it be that in our experiment most participants seem to use a negative stepsize in the dynamic war of attrition treatment? To understand this better let us compare the equilibrium bidding functions in the war of attrition and the allpay auction. Figure 11 shows examples for the equilibrium bidding functions in the war of attrition (equation (4)) and the allpay auction (equation (10)) as solid lines. Since Ð Ñ c c β W e (c) = the function is not easy to approximate with a stepwise linear function. The dotted line shows the result of an OLS regression of a random sample of cost values. We see that, since the left part of the function is very steep, we obtain a negative stepsize. The equilibrium bidding function for 24
26 the allpay auction is much easier to fit with a stepwise linear function (the dotted line is very close to the equilibrium bidding function). Any step of a stepwise linear approximation will be small. 5 Concluding remarks In this series of experiments we address a couple of hypotheses, some related to theoretical studies of the war of attrition, some to other experiments. In line with previous experiments we have found overbidding in allpay auctions. However, we have seen that overbidding does not carry over to wars of attrition. While equilibrium expenditures are the same in allpay auctions and in wars of attrition we have found that in the experiment expenditures are lower in wars of attrition than in allpay auctions. We could see that two effects work handinhand: First, moving from a firstprice format (as in the allpay auction) to a secondprice format (as in the static version of the war of attrition) reduces expenditures. Second, moving from a static format (as in the static version of the war of attrition) to a dynamic format (as in the proper war of attrition) lowers expenditures even more. These results might be interesting for the designer of a contest. If the designer of a contest wants to maximise expenditure (as, e.g., in a competition of students) it might be attractive to choose a firstprice format. Contestants should be able to commit to a bid and no information about the number of remaining contestants should be made public. If expenditures are socially wasteful (as in a patent race) we should do the opposite. In line with previous findings of Müller and Schotter (2007) we have found support for discontinuous individual bidding functions in the allpay auction. For bids in allpay auctions it matters mainly whether costs are high or low. Participants do not distinguish between intermediate values. With low costs bid are high (and are not much affected by the exact level of the costs), with high costs bids are low (and are, again, not much affected by the exact level of the costs). Also in the war of attrition bidders have a tendency to follow two different regimes, one for low and one for high costs. However, the very simple approximation, which worked 25
ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationCournot s model of oligopoly
Cournot s model of oligopoly Single good produced by n firms Cost to firm i of producing q i units: C i (q i ), where C i is nonnegative and increasing If firms total output is Q then market price is P(Q),
More informationIntroduction to Auctions
Introduction to Auctions Lots of good theory and empirical work. game is simple with wellspecified rules actions are observed directly payoffs can sometimes be inferred Also, lots of data. government
More informationLobbying contests with endogenous policy proposals
Lobbying contests with endogenous policy proposals Johannes Münster March 31, 2004 Abstract Lobbyists choose what to lobby for. This endogeneity of policy proposals has recently been modelled by Epstein
More informationHybrid Auctions Revisited
Hybrid Auctions Revisited Dan Levin and Lixin Ye, Abstract We examine hybrid auctions with affiliated private values and riskaverse bidders, and show that the optimal hybrid auction trades off the benefit
More informationBayesian Nash Equilibrium
. Bayesian Nash Equilibrium . In the final two weeks: Goals Understand what a game of incomplete information (Bayesian game) is Understand how to model static Bayesian games Be able to apply Bayes Nash
More informationInternet Advertising and the Generalized Second Price Auction:
Internet Advertising and the Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords Ben Edelman, Harvard Michael Ostrovsky, Stanford GSB Michael Schwarz, Yahoo! Research A Few
More informationThe Role of Money Illusion in Nominal Price Adjustment
The Role of Money Illusion in Nominal Price Adjustment LUBA PETERSEN AND ABEL WINN Online Appendices 1 Appendix A. Adaptive Best Responding in FT s Experiments and Our Revised Experiments Section 1. Statistical
More informationJar of Coins. Jar contains some amount of coins
Jar of Coins Jar contains some amount of coins Jar of Coins Jar contains some amount of coins Write down a bid on the jar Jar of Coins Jar contains some amount of coins Write down a bid on the jar Is this
More informationChapter 11: Game Theory & Competitive Strategy
Chapter 11: Game Theory & Competitive Strategy Game theory helps to understand situations in which decisionmakers strategically interact. A game in the everyday sense is: A competitive activity... according
More informationUniversidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games
Session 1: 1, 2, 3, 4 Session 2: 5, 6, 8, 9 Universidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games 1. Consider the following game in the normal form: Player 1 Player
More informationManagerial Economics
Managerial Economics Unit 8: Auctions Rudolf WinterEbmer Johannes Kepler University Linz Winter Term 2012 Managerial Economics: Unit 8  Auctions 1 / 40 Objectives Explain how managers can apply game
More informationAllpay auctions an experimental study
Journal of Economic Behavior & Organization Vol. 61 (2006) 255 275 Allpay auctions an experimental study Uri Gneezy a,, Rann Smorodinsky b,1 a The University of Chicago, Graduate School of Business, 1101
More informationChapter 3 : Dynamic Game Theory.
Chapter 3 : Dynamic Game Theory. In the previous chapter we focused on static games. However for many important economic applications we need to think of the game as being played over a number of time
More informationTest of Hypotheses. Since the NeymanPearson approach involves two statistical hypotheses, one has to decide which one
Test of Hypotheses Hypothesis, Test Statistic, and Rejection Region Imagine that you play a repeated Bernoulli game: you win $1 if head and lose $1 if tail. After 10 plays, you lost $2 in net (4 heads
More information1. Suppose demand for a monopolist s product is given by P = 300 6Q
Solution for June, Micro Part A Each of the following questions is worth 5 marks. 1. Suppose demand for a monopolist s product is given by P = 300 6Q while the monopolist s marginal cost is given by MC
More informationSealed Bid Second Price Auctions with Discrete Bidding
Sealed Bid Second Price Auctions with Discrete Bidding Timothy Mathews and Abhijit Sengupta August 16, 2006 Abstract A single item is sold to two bidders by way of a sealed bid second price auction in
More informationAdaptation And Learning In Blotto Games
Adaptation And Learning In Blotto Games Elais Jackson 1 1 Department of History, Geography, and Political Science, Tennessee State University 20100811 Abstract When Game Theory began to take off, the
More informationMidterm Exam. Economics Spring 2014 Professor Jeff Ely
Midterm Exam Economics 4103 Spring 014 Professor Jeff Ely Instructions: There are three questions with equal weight. You have until 10:50 to complete the exam. This is a closedbook and closednotebook
More information4 Eliminating Dominated Strategies
4 Eliminating Dominated Strategies Um so schlimmer für die Tatsache (So much the worse for the facts) Georg Wilhelm Friedrich Hegel 4.1 Dominated Strategies Suppose S i is a finite set of pure strategies
More informationNext Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due
Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 6 Sept 25 2007 Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due Today: the pricediscriminating
More informationOLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique  ie the estimator has the smallest variance
Lecture 5: Hypothesis Testing What we know now: OLS is not only unbiased it is also the most precise (efficient) unbiased estimation technique  ie the estimator has the smallest variance (if the GaussMarkov
More informationMidsummer Examinations 2013
[EC 7089] Midsummer Examinations 2013 No. of Pages: 5 No. of Questions: 5 Subject [Economics] Title of Paper [EC 7089: GAME THEORY] Time Allowed [2 hours (TWO HOURS)] Instructions to candidates Answer
More informationExam #2 cheat sheet. The space provided below each question should be sufficient for your answer. If you need additional space, use additional paper.
Exam #2 cheat sheet The space provided below each question should be sufficient for your answer. If you need additional space, use additional paper. You are allowed to use a calculator, but only the basic
More information, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (
Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we
More informationOligopoly and Strategic Pricing
R.E.Marks 1998 Oligopoly 1 R.E.Marks 1998 Oligopoly Oligopoly and Strategic Pricing In this section we consider how firms compete when there are few sellers an oligopolistic market (from the Greek). Small
More informationCALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 15 scale to 0100 scores When you look at your report, you will notice that the scores are reported on a 0100 scale, even though respondents
More informationOptimal Auctions Continued
Lecture 6 Optimal Auctions Continued 1 Recap Last week, we... Set up the Myerson auction environment: n riskneutral bidders independent types t i F i with support [, b i ] residual valuation of t 0 for
More informationGames of Incomplete Information
Games of Incomplete Information Jonathan Levin February 00 Introduction We now start to explore models of incomplete information. Informally, a game of incomplete information is a game where the players
More informationChapter 6. Noncompetitive Markets 6.1 SIMPLE MONOPOLY IN THE COMMODITY MARKET
Chapter 6 We recall that perfect competition was theorised as a market structure where both consumers and firms were price takers. The behaviour of the firm in such circumstances was described in the Chapter
More informationAn Empirical Analysis of Insider Rates vs. Outsider Rates in Bank Lending
An Empirical Analysis of Insider Rates vs. Outsider Rates in Bank Lending Lamont Black* Indiana University Federal Reserve Board of Governors November 2006 ABSTRACT: This paper analyzes empirically the
More informationWeek 7  Game Theory and Industrial Organisation
Week 7  Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number
More informationComputational Learning Theory Spring Semester, 2003/4. Lecture 1: March 2
Computational Learning Theory Spring Semester, 2003/4 Lecture 1: March 2 Lecturer: Yishay Mansour Scribe: Gur Yaari, Idan Szpektor 1.1 Introduction Several fields in computer science and economics are
More informationWeek 3: Monopoly and Duopoly
EC202, University of Warwick, Term 2 1 of 34 Week 3: Monopoly and Duopoly Dr Daniel Sgroi Reading: 1. Osborne Sections 3.1 and 3.2; 2. Snyder & Nicholson, chapters 14 and 15; 3. Sydsæter & Hammond, Essential
More informationRevenue Equivalence Revisited
Discussion Paper No. 175 Revenue Equivalence Revisited Radosveta IvanovaStenzel* Timothy C. Salmon** October 2006 *Radosveta IvanovaStenzel, HumboldtUniversity of Berlin, Department of Economics, Spandauer
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitelyrepeated prisoner s dilemma
More informationCHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY
CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY EXERCISES 3. Two computer firms, A and B, are planning to market network systems for office information management. Each firm can develop either a fast,
More information6.207/14.15: Networks Lectures 1921: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning
6.207/14.15: Networks Lectures 1921: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning Daron Acemoglu and Asu Ozdaglar MIT November 23, 25 and 30, 2009 1 Introduction
More informationA note on stochastic public revelation and voluntary contributions to public goods
July 2014 A note on stochastic public revelation and voluntary contributions to public goods JOHN M. SPRAGGON * LUCIA ANDREA VERGARA SOBARZO JOHN K. STRANLUND Department of Resource Economics University
More informationGames of Incomplete Information
Games of Incomplete Information Yan Chen November 16, 2005 Games of Incomplete Information Auction Experiments Auction Theory Administrative stuff Games of Incomplete Information Several basic concepts:
More informationThe Sealed Bid Auction Experiment:
The Sealed Bid Auction Experiment: This is an experiment in the economics of decision making. The instructions are simple, and if you follow them carefully and make good decisions, you may earn a considerable
More information8 Modeling network traffic using game theory
8 Modeling network traffic using game theory Network represented as a weighted graph; each edge has a designated travel time that may depend on the amount of traffic it contains (some edges sensitive to
More informationHow to Win the Stock Market Game
How to Win the Stock Market Game 1 Developing ShortTerm Stock Trading Strategies by Vladimir Daragan PART 1 Table of Contents 1. Introduction 2. Comparison of trading strategies 3. Return per trade 4.
More informationA Note on Multiwinner Contest Mechanisms
A Note on Multiwinner Contest Mechanisms Subhasish M. Chowdhury and SangHyun Kim February 20, 2014 Abstract We consider a multiwinner nested elimination contest in which losers are sequentially eliminated
More informationPASS Sample Size Software. Linear Regression
Chapter 855 Introduction Linear regression is a commonly used procedure in statistical analysis. One of the main objectives in linear regression analysis is to test hypotheses about the slope (sometimes
More informationChapter 9. Auctions. 9.1 Types of Auctions
From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint online at http://www.cs.cornell.edu/home/kleinber/networksbook/
More informationCompetitive markets as twoplayer games
Competitive markets as twoplayer games Antonio Quesada Departament d Economia, Universitat Rovira i Virgili, Avinguda de la Universitat 1, 43204 Reus, Spain 23rd December 2008 126.14 Abstract With the
More informationJournal Of Business And Economics Research Volume 1, Number 2
Bidder Experience In Online Auctions: Effects On Bidding Choices And Revenue Sidne G. Ward, (Email: wards@umkc.edu), University of Missouri Kansas City John M. Clark, (Email: clarkjm@umkc.edu), University
More informationA dynamic auction for multiobject procurement under a hard budget constraint
A dynamic auction for multiobject procurement under a hard budget constraint Ludwig Ensthaler Humboldt University at Berlin DIW Berlin Thomas Giebe Humboldt University at Berlin March 3, 2010 Abstract
More informationgovernments and firms, and in particular ebay on market design; the views expressed are our own.
On the Design of Simple Multiunit Online Auctions Thomas Kittsteiner (London School of Economics) and Axel Ockenfels (University of Cologne) 1 The increased use of online market places (like ebay) by
More informationChapter 10. Key Ideas Correlation, Correlation Coefficient (r),
Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables
More informationUnit 8. Firm behaviour and market structure: monopolistic competition and oligopoly
Unit 8. Firm behaviour and market structure: monopolistic competition and oligopoly Learning objectives: to understand the interdependency of firms and their tendency to collude or to form a cartel; to
More informationUnit 7. Firm behaviour and market structure: monopoly
Unit 7. Firm behaviour and market structure: monopoly Learning objectives: to identify and examine the sources of monopoly power; to understand the relationship between a monopolist s demand curve and
More informationREGRESSION LINES IN STATA
REGRESSION LINES IN STATA THOMAS ELLIOTT 1. Introduction to Regression Regression analysis is about eploring linear relationships between a dependent variable and one or more independent variables. Regression
More informationECON 40050 Game Theory Exam 1  Answer Key. 4) All exams must be turned in by 1:45 pm. No extensions will be granted.
1 ECON 40050 Game Theory Exam 1  Answer Key Instructions: 1) You may use a pen or pencil, a handheld nonprogrammable calculator, and a ruler. No other materials may be at or near your desk. Books, coats,
More informationLECTURE 10: GAMES IN EXTENSIVE FORM
LECTURE 10: GAMES IN EXTENSIVE FORM Sequential Move Games 2 so far, we have only dealt with simultaneous games (players make the decisions at the same time, or simply without knowing what the action of
More informationSimple Linear Regression Chapter 11
Simple Linear Regression Chapter 11 Rationale Frequently decisionmaking situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related
More informationThe Shapley Value and the Core
The Shapley Value and the Core Lecture 23 The Shapley Value and the Core Lecture 23, Slide 1 Lecture Overview 1 Recap 2 Analyzing Coalitional Games 3 The Shapley Value 4 The Core The Shapley Value and
More informationBidding with Securities: ProfitShare (Equity) vs. Cash Auctions
Elmar G. Wolfstetter 1/13 Bidding with Securities: ProfitShare (Equity) vs. Cash Auctions Elmar G. Wolfstetter Elmar G. Wolfstetter 2/13 Outline 1 Introduction 2 Equilibrium 3 (Almost) full surplus extraction
More informationThe Behavioral Economics of Incentives
The Behavioral Economics of Incentives The Role of Sanctioning Threats & Loss Aversion Ernst Fehr, Armin Falk, Lorenz Götte Ernst Fehr Experimental & Behavioral Economics Trust & Sanctioning Threats (Fehr
More informationOverview: Auctions and Bidding. Examples of Auctions
Overview: Auctions and Bidding Introduction to Auctions Openoutcry: ascending, descending Sealedbid: firstprice, secondprice Private Value Auctions Common Value Auctions Winner s curse Auction design
More information3.7 Evolutionary Game Theory
3.7 Evolutionary Game Theory The HawkDove game is used in biology to explain why aggression is present within populations, but is not always observed. Consider the symmetric HawkDove game. The intuition
More informationTOPIC 4: ASYMMETRIC INFORMATION MODELS OF CAPITAL STRUCTURE
TOPIC 4: ASYMMETRIC INFORMATION MODELS OF CAPITAL STRUCTURE 1. Introduction The recent economic literature has emphasized a lot the importance of asymmetric information to our understanding of a wide range
More informationGame Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions
Game Theory and Algorithms Lecture 0: Extensive Games: Critiques and Extensions March 3, 0 Summary: We discuss a game called the centipede game, a simple extensive game where the prediction made by backwards
More information6.207/14.15: Networks Lecture 16: Cooperation and Trust in Networks
6.207/14.15: Networks Lecture 16: Cooperation and Trust in Networks Daron Acemoglu and Asu Ozdaglar MIT November 4, 2009 1 Introduction Outline The role of networks in cooperation A model of social norms
More informationEconomics 200C, Spring 2012 Practice Midterm Answer Notes
Economics 200C, Spring 2012 Practice Midterm Answer Notes I tried to write questions that are in the same form, the same length, and the same difficulty as the actual exam questions I failed I think that
More informationThe Limits of Price Discrimination
The Limits of Price Discrimination Dirk Bergemann, Ben Brooks and Stephen Morris University of Zurich May 204 Introduction: A classic economic issue... a classic issue in the analysis of monpoly is the
More informationUCLA. Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory
UCLA Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.
More information6.254 : Game Theory with Engineering Applications Lecture 1: Introduction
6.254 : Game Theory with Engineering Applications Lecture 1: Introduction Asu Ozdaglar MIT February 2, 2010 1 Introduction Optimization Theory: Optimize a single objective over a decision variable x R
More informationA Simple Model of Price Dispersion *
Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion
More informationPredatory Pricing: Rare Like a Unicorn?
Predatory Pricing: Rare Like a Unicorn? Rosario Gomez, Jacob K. Goeree, and Charles A. Holt * I. INTRODUCTION Despite the discovery of predatory intent in several widely cited antitrust cases, many industrial
More informationSECTION A (Multiple choice)
MIDSUMMER EXAMINATIONS 2009 Subject Title of Paper Time Allowed ECONOMICS : MARKET POWER AND MARKET FAILURE Two Hours (2 hours) Instructions to candidates Candidates should attempt ALL of Section A, ONE
More informationSTATISTICA Formula Guide: Logistic Regression. Table of Contents
: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 SigmaRestricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary
More informationStructural Econometric Modeling in Industrial Organization Handout 1
Structural Econometric Modeling in Industrial Organization Handout 1 Professor Matthijs Wildenbeest 16 May 2011 1 Reading Peter C. Reiss and Frank A. Wolak A. Structural Econometric Modeling: Rationales
More informationUniversity of Kent. School of Economics Discussion Papers
University of Kent School of Economics Discussion Papers Endogenous Leadership in a Coordination Game with Conflict of Interest and Asymmetric Information Edward Cartwright, Joris Gillet, Mark Van Vugt
More information10.1 Bipartite Graphs and Perfect Matchings
10.1 Bipartite Graphs and Perfect Matchings In bipartite graph the nodes are divided into two categories and each edge connects a node in one category to a node in the other category. For example: Administrators
More information6.254 : Game Theory with Engineering Applications Lecture 10: Evolution and Learning in Games
6.254 : Game Theory with Engineering Applications Lecture 10: and Learning in Games Asu Ozdaglar MIT March 9, 2010 1 Introduction Outline Myopic and Rule of Thumb Behavior arily Stable Strategies Replicator
More informationSequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium
Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses
More informationEconomic Insight from Internet Auctions. Patrick Bajari Ali Hortacsu
Economic Insight from Internet Auctions Patrick Bajari Ali Hortacsu ECommerce and Internet Auctions Background $45 billion in ecommerce in 2002 632 million items listed for auction on ebay alone generating
More informationAN INTRODUCTION TO GAME THEORY
AN INTRODUCTION TO GAME THEORY 2008 AGIInformation Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. MARTIN J. OSBORNE University of Toronto
More informationMATH MODULE 11. Maximizing Total Net Benefit. 1. Discussion M111
MATH MODULE 11 Maximizing Total Net Benefit 1. Discussion In one sense, this Module is the culminating module of this Basic Mathematics Review. In another sense, it is the starting point for all of the
More informationMonopolistic Competition, Oligopoly, and maybe some Game Theory
Monopolistic Competition, Oligopoly, and maybe some Game Theory Now that we have considered the extremes in market structure in the form of perfect competition and monopoly, we turn to market structures
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationAuctions. Economics Auction Theory. Instructor: Songzi Du. Simon Fraser University. September 15, 2016
Auctions Economics 383  Auction Theory Instructor: Songzi Du Simon Fraser University September 15, 2016 ECON 383 (SFU) Auctions September 15, 2016 1 / 28 Auctions Mechanisms of transaction: bargaining,
More informationThe Role of Rest in the NBA HomeCourt Advantage
The Role of Rest in the NBA HomeCourt Advantage Oliver Entine Department of Statistics, The Wharton School of the University of Pennsylvania, 3730 Walnut St., Philadelphia, PA 19104, U.S.A. entine4@wharton.upenn.edu
More informationDo Commodity Price Spikes Cause LongTerm Inflation?
No. 111 Do Commodity Price Spikes Cause LongTerm Inflation? Geoffrey M.B. Tootell Abstract: This public policy brief examines the relationship between trend inflation and commodity price increases and
More informationThe behaviour of players in games with a mixed strategy Nash equilibrium: Evidence from a stylised Poker experiment.
The behaviour of players in games with a mixed strategy Nash equilibrium: Evidence from a stylised Poker experiment. Department of Economics, University of Essex Katherine Drakeford Registration Number:
More informationAdvanced High School Statistics. Preliminary Edition
Chapter 2 Summarizing Data After collecting data, the next stage in the investigative process is to summarize the data. Graphical displays allow us to visualize and better understand the important features
More informationOnline Appendix for AngloDutch Premium Auctions in EighteenthCentury Amsterdam
Online Appendi for AngloDutch Premium Auctions in EighteenthCentury Amsterdam Christiaan van Bochove, Lars Boerner, Daniel Quint January, 06 The first part of Theorem states that for any, ρ > 0, and
More informationLATE AND MULTIPLE BIDDING IN SECOND PRICE INTERNET AUCTIONS: THEORY AND EVIDENCE CONCERNING DIFFERENT RULES FOR ENDING AN AUCTION
LATE AND MULTIPLE BIDDING IN SECOND PRICE INTERNET AUCTIONS: THEORY AND EVIDENCE CONCERNING DIFFERENT RULES FOR ENDING AN AUCTION AXEL OCKENFELS ALVIN E. ROTH CESIFO WORKING PAPER NO. 992 CATEGORY 9: INDUSTRIAL
More informationIn this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing.
3.0  Chapter Introduction In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing. Categories of Statistics. Statistics is
More informationOligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry s output.
Topic 8 Chapter 13 Oligopoly and Monopolistic Competition Econ 203 Topic 8 page 1 Oligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry
More informationAdvanced Microeconomics
Advanced Microeconomics Static Bayesian games Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics / 7 Part E. Bayesian games and mechanism design Static Bayesian
More informationWhy is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013)
Why is Insurance Good? An Example Jon Bakija, Williams College (Revised October 2013) Introduction The United States government is, to a rough approximation, an insurance company with an army. 1 That is
More informationGame Theory. UCI Math Circle, Fall 2015
Game Theory UCI Math Circle, Fall 2015 (1) (Prisoner s Dilemma) Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with
More informationEquilibrium: Illustrations
Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23. Martin.Osborne@utoronto.ca http://www.economics.utoronto.ca/osborne Copyright 1995 2002 by Martin J. Osborne.
More informationAuctions: Experiments (for New Palgrave Dictionary of Economics) John H. Kagel and Dan Levin Ohio State University
Auctions: Experiments (for New Palgrave Dictionary of Economics) John H. Kagel and Dan Levin Ohio State University Abstract: Experiments permit rigorous testing of auction theory. In singleunit private
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More informationWhat is Evolutionary Game Theory?
What is Evolutionary Game Theory? Origins: Genetics and Biology Explain strategic aspects in evolution of species due to the possibility that individual fitness may depend on population frequency Basic
More information